Non-Perturbative Analysis of Physical and Mathematical Models

在对自然界进行建模时，科学家经常必须依赖于导致关于正在研究的系统的不完整信息的近似。基本粒子物理、统计力学和万有引力都是如此。因此，一个根本的问题是如何从有限的数据中提取尽可能多的信息，并尽可能地保持最高的准确性和置信度。这个项目将利用数学的几个领域来分析数学和物理中的这个问题。此外，研究人员的目标是进一步发展激光与物质相互作用的数学。该项目为研究生提供了培训和研究机会。该项目涉及的一个中心主题是物质和辐射的相互作用。对于时间相关的激光诱导光电效应，研究人员将把一种建设性的数学技术应用于实验中使用的光电离化模型和双频模型，以微调光电输出，并在与实验者的交叉学科工作中，预测和优化过程。该项目的第二个目标是在只能计算有限数量的微扰展开项的情况下，以尽可能高的精度和置信度来确定微扰理论范围之外的物理和数学模型的行为。研究人员的目标是开发出在特定数学条件下最优的、超出以前达到的精度的方法。这些方法也将应用于可以获得精确解的数学问题，但它们的数值利用是计算上的需求。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In modeling the natural world, scientists often must rely on approximations that result in incomplete information about a system under study. Such is the case in elementary particle physics, statistical mechanics, and gravitation. A fundamental question is then how to extract from limited data as much information as possible, with the possible highest level of accuracy and confidence. This project will draw on several areas of mathematics to analyze this issue in mathematics and physics. Additionally, the investigator aims to further develop the mathematics of laser-matter interaction. The project offers training research opportunities for graduate students. A central topic addressed in the project is the interaction of matter and radiation. For the time-dependent laser-induced photoelectric effect, the investigator will apply a constructive mathematical technique to the photoionization models and the two-frequency models used in experiments, to fine-tune the photoelectric output and, in interdisciplinary work with experimentalists, to predict and optimize the processes. A second aim of the project is to determine the behavior of physical and mathematical models outside the range of perturbation theory, with the highest possible level of accuracy and confidence, when only a limited number of terms of a perturbation expansion can be calculated. The investigator aims to develop methods that exceed the precision previously achieved and that are optimal under specific mathematical conditions. The methods will be applied also to mathematical problems in which exact solutions are available, but their numerical utilization is computationally demanding.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.